무게중심과 로그

삼각형 $\mathrm{ABC}$의 무게중심의 좌표는 $\displaystyle \left( \frac{0 + 2 a + \left( - \log \mathit{_{2}} 9 \right)}{3} , \frac{\left( - \log \mathit{_{2}} 9 \right) + \log \mathit{_{2}} 7 + a}{3} \right)$ $\displaystyle \frac{0 + 2 a + \left( - \log \mathit{_{2}} 9 \right)}{3} = b$, $2 a - \log \mathit{_{2}} 9 = 3 b$ $\displaystyle b = \frac{2}{3} a - \frac{1}{3} \log \mathit{_{2}} 9$ $\cdots$$\cdots$ ㉠ $\displaystyle \frac{\left( - \log \mathit{_{2}} 9 \right) + \log \mathit{_{2}} 7 + a}{3} = \log \mathit{_{8}} 7 = \frac{1}{3} \log \mathit{_{2}} 7$ $- \log \mathit{_{2}} 9 + \log \mathit{_{2}} 7 + a = \log \mathit{_{2}} 7$ $a = \log \mathit{_{2}} 9$ $\cdots$$\cdots$ ㉡ 두 식 ㉠, ㉡을 연립하면 $\displaystyle b = \frac{2}{3} \log \mathit{_{2}} 9 - \frac{1}{3} \log \mathit{_{2}} 9 = \frac{1}{3} \log \mathit{_{2}} 9$ $a + 3 b = \log \mathit{_{2}} 9 + \log \mathit{_{2}} 9 = \log \mathit{_{2}} 81$ $2 ^{a + 3 b} = 2 ^{\log \mathit{_{2}} 81}$ $2 ^{\log \mathit{_{2}} 81} = k$라 하면 $\log \mathit{_{2}} k = \log \mathit{_{2}} 81$, $k = 81$ 따라서 $2 ^{a + 3 b} = 81$